Variational methods in materials science

Three problems are being investigated in this thesis. The first two relate to the modelling and analysis of martensitic phase transitions, while the third is concerned with some mathematical tools used in this setting. After a short introduction (Chapter 1) and overviews of the calculus of variations and martensitic phase transformations (Chapter 2), the research part of this thesis is divided into three chapters. We show in Chapter 3 that for the two wells $\mathrm{SO}(3)U$ and $\mathrm{SO}(3)V$ to be rank-one connected, where the $3\times 3$ symmetric positive definite $U$ and $V$ have the same eigenvalues, it is necessary and sufficient that $\mathrm{det}(U-V)=0$, a result that does not hold in higher dimensions. Using this criterion and a result of Gurtin, formulae for the twinning plane and the shearing vector are obtained, which yield an extremely simple condition for the occurrence of so-called compound twins. Our results also provide a simple classification of the twinning mode of the two wells by looking at the crystallographic properties of the eigenvectors of the difference $U-V$. As an illustration, we apply our results to cubic-to-tetra gonal,tetragonal-to-monoclinic and cubic-to-monoclinic transitions. Chapter 4 focuses on the mathematical analysis of biaxial loading experiments in martensite, more particularly on how hysteresis relates to metastability. These experiments were carried out by Chu and James and their mathematical treatment was initiated by Ball, Chu and James. Experimentally it is observed that a homogeneous deformation $y_1(x)= U_1x$ is the stable state for `small' loads while $y_2(x)=U_2x$ is stable for `large' loads. A model was proposed by Ball, Chu and James which, for a certain intermediate range of loads, predicts crucially that $y_1(x)=U_1x$ remains metastable i.e., a local - as opposed to global - minimiser of the energy). This result explains convincingly the hysteresis that is observed experimentally. It is easy to get an upper bound for when metastability finishes. However, it was also noticed that this bound (the Schmid Law) may not be sharp, though this required some geometric conditions on the sample. In this chapter, we rigorously justify the Ball-Chu-James model by means of De Giorgi's $\Gamma$-convergence, establish some properties of local minimisers of the (limiting) energy and prove the metastability result mentioned above. An important part of the chapter is then devoted to establishing which geometric conditions are necessary and sufficient for the counter-example to the Schmid Law to apply. Finally, Chapter 5 investigates the structure of the solutions to the two-well problem. Restricting ourselves to the subset $K=\{H\}\cup \mathrm{SO}(2)V \subset\mathrm{SO}(2)U\cup\mathrm{SO}(2)V$ and assuming the two wells to be compatible, we let $T_1$ and $T_2$ denote the two (not necessarily distinct) twins of $H$ on $\mathrm{SO}(2)V$ and ask the following question: if $\nu_x$ is a non-trivial gradient Young measure almost everywhere supported on $K$, does its support necessarily contain a pair of rank-one connected matrices on a set of positive measure? Although we do not provide a solution for the general case, we show that this is true whenever (a) $\nu_x\equiv \nu$ is homogeneous and $\mathrm{supp}\nu\cap \mathrm{SO}(2)V$ is connected, (b) $\nu_x\equiv \nu$ is homogeneous and $T_1=T_2$ i.e., when the two wells are trivially rank-one connected) or (c) $\mathrm{supp}\nu_x \subset F$ a.e., for some finite set $F$. We also establish a more general case provided a strong `rigidity' conjecture holds

Incompatible sets of gradients and metastability

We give a mathematical analysis of a concept of metastability induced by incompatibility. The physical setting is a single parent phase, just about to undergo transformation to a product phase of lower energy density. Under certain conditions of incompatibility of the energy wells of this energy density, we show that the parent phase is metastable in a strong sense, namely it is a local minimizer of the free energy in an $L^1$ neighbourhood of its deformation. The reason behind this result is that, due to the incompatibility of the energy wells, a small nucleus of the product phase is necessarily accompanied by a stressed transition layer whose energetic cost exceeds the energy lowering capacity of the nucleus. We define and characterize incompatible sets of matrices, in terms of which the transition layer estimate at the heart of the proof of metastability is expressed. Finally we discuss connections with experiment and place this concept of metastability in the wider context of recent theoretical and experimental research on metastability and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea

Radio-frequency ablation as primary management of well-tolerated sustained monomorphic ventricular tachycardia in patients with structural heart disease and left ventricular ejection fraction over 30%.

AIMS: Patients with well-tolerated sustained monomorphic ventricular tachycardia (SMVT) and left ventricular ejection fraction (LVEF) over 30% may benefit from a primary strategy of VT ablation without immediate need for a 'back-up' implantable cardioverter-defibrillator (ICD). METHODS AND RESULTS: One hundred and sixty-six patients with structural heart disease (SHD), LVEF over 30%, and well-tolerated SMVT (no syncope) underwent primary radiofrequency ablation without ICD implantation at eight European centres. There were 139 men (84%) with mean age 62 ± 15 years and mean LVEF of 50 ± 10%. Fifty-five percent had ischaemic heart disease, 19% non-ischaemic cardiomyopathy, and 12% arrhythmogenic right ventricular cardiomyopathy. Three hundred seventy-eight similar patients were implanted with an ICD during the same period and serve as a control group. All-cause mortality was 12% (20 patients) over a mean follow-up of 32 ± 27 months. Eight patients (40%) died from non-cardiovascular causes, 8 (40%) died from non-arrhythmic cardiovascular causes, and 4 (20%) died suddenly (SD) (2.4% of the population). All-cause mortality in the control group was 12%. Twenty-seven patients (16%) had a non-fatal recurrence at a median time of 5 months, while 20 patients (12%) required an ICD, of whom 4 died (20%). CONCLUSION: Patients with well-tolerated SMVT, SHD, and LVEF > 30% undergoing primary VT ablation without a back-up ICD had a very low rate of arrhythmic death and recurrences were generally non-fatal. These data would support a randomized clinical trial comparing this approach with others incorporating implantation of an ICD as a primary strategy

Tracking of Stepwise Ablation of Persistent Atrial Fibrillation using Synchronization of nearby Electrogams

Intracardiac organization indices such as atrial fibril- lation (AF) cycle length (AFCL) have been used to track the efficiency of stepwise catheter ablation (step-CA) of long-standing persistent AF (pers-AF), however, with lim- ited success. The timing between nearby bipolar intracar- diac electrograms (EGMs) reflects the spatial dynamics of wavelets during AF. The extent of synchronization between EGMs is an indirect measure of AF spatial organization. The synchronization between nearby EGMs during step- CA of pers-AF was evaluated using new indices based on the cross-correlation. The first one (spar(W)) quantifies the sparseness of the cross-correlation of local activation times. The second one (OI(W)) reflects the local concen- tration around the largest peak of the cross-correlation. By computing their relative evolution during step-CA until AF termination (AF-term), we found that OI(W) appeared su- perior to AFCL and spar(W) to track the effect of step-CA "en route" to AF-term

High bi-atrial organization in patients with long-standing persistent atrial fibrillation terminated within the left atrium

Sustained atrial fibrillation (AF) is maintained by sites displaying high dominant frequency (DF). In patients (pts) with long-standing persistent AF (LS-pAF), their spatial distribution and the presence of a left-to-right atrial DF gradient remain poorly known. We hypothesized that the pre-ablation bi-atrial frequency characteristics of LS-pAF pts terminated within the left atrium (LT) are different from that of non terminated (NT) ones. Methods: 23 consecutive pts (59±7y, LS-pAF duration 19±12m) underwent stepwise catheter ablation (step-CA) consisting in pulmonary veins isolation, left atrial (LA) defragmentation, and right atrial (RA) ablations for non terminated AF. A quadripolar catheter (CAT) was placed into the RA appendage (RAA), a decapolar CAT into the coronary sinus (CS) and a duodecapolar CAT into the LA divided into 8 segments. For each segment, 20-sec of bipolar recording was acquired. The DF was defined as the largest peak in the power spectrum (3-15 Hz). The inter-atrial DF gradient was defined as the DF difference between LA and RA appendages. Results: LS-pAF was terminated in 83% (19/23) of the pts: 17 LT, 2 during RA ablation and 4 NT. The figure shows that before ablation bi-atrial DF values of LT pts are significantly lower than that of NT pts for each LA segment as well as for the RAA (p < 0.05). No significant LA-to-RA DF gradient was observed both for LT (0.3±0.5 Hz, p=ns) and NT (0.5±0.03 Hz, p=ns) pts. No significant difference in DF values was observed between LA segments. Conclusions: The lower DF of LT pts is suggestive of a higher organization within both atria compared to NT pts. Our findings suggest that low bi-atrial DF values, but not inter-atrial DF gradient, might be of interest for selecting LS-pAF candidates for sinus rhythm restoration by step-CA

A New Method for ECG Tracking of Persistent Atrial Fibrillation Termination during Stepwise Ablation

Stepwise radiofrequency catheter ablation (step-CA) has become the treatment of choice for the restoration of sinus rhythm (SR) in patients with long-standing persistent atrial fibrillation (pers-AF). Its success rate appears limited as the amount of ablation to achieve long term SR is unknown. Multiple organization indexes (OIs) have been previously developed to track the organization of AF during step-CA, however, with limited success. We report an adaptive method for tracking AF termination (AF-term) based on OIs characterizing the relationship between harmonic components of atrial activity from the surface ECG of AF activity. By computing their relative evolution during the last two steps preceding AF-term, we found that the performance of our OIs was superior to classical indices to track the efficiency of step-CA "en route" to AF-term. Our preliminary results suggest that the gradual synchronization between the fundamental and its first harmonic of AF activity appears as a promising parameter for predicting AF-term during step-CA